3x+4y=24. What is the smallest integer solution for x+y?

Since we are looking for integer solutions, x can get really small, as long as it remains a multiple of 12. This ensures that y can get it back to 24. Therefore, the answer is - infinity.

lets solve for x, and then we'll find an integer solution:

3x + 4y = 24

3x = 24 - 4y

x = (24 - 4y) / 3

so the right side has to be an integer, or 24 - 4y has to be a multiple of 3:

24 - 4y

then, y has to be 3, see:

x = (24 - 4y) / 3

x = (24 - 4 (3)) / 3

x = (24 - 12) / 3

x = (12) / 3

x = 4

then, we have x = 4 and y = 3